Calendar

The calendar is a way to reckon time, specifically the passage of days in relation to months and years. The choice of calendar determines when the anniversaries of events will take place. Hence, it determines when and how long their celebrations—the feasts and fasts—will be, and even whether those take place at all.

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Historical origins

Observational calendars

The simplest types of calendar involve simply keeping track of the passage of time after it has already occurred. This is always correct by definition; a lunar calendar which relies on physically looking up to confirm that the new moon has occurred will have that event on the right date. Such methods have historically been used and are still in use by some, for example the Islamic month of Ramadan is still determined by observation in spite of our ability to calculate the phases of the moon.

However, in order to plan ahead one must have some way of extrapolating to future dates. This is critical for planning the feasts of the Church—without knowing the date of the Paschal new moon, we would not know when to begin lent, for example.

The church of Alexandria was historically entrusted with the responsibility of calculating the date of Pascha as it was a center of astronomical learning.

The Church calendar

Main article: Church calendar.

The Julian calendar

The Julian calendar was the Romans' improved solar calendar. With every fourth year being a leap year, it tracks the tropical year almost perfectly while being simple to calculate. The months of the Julian calendar, however, do not approximate lunar months very much at all.

Given that the Roman Empire was the major civil power within whose territory the Church had much of her flock, it was only natural that the Church would also use this calendar for celebrating the anniversaries of her martyrs' victories. Hence the feasts of the saints were recorded in the menaion according to Julian date, with Pascha according to a luni-solar calculation discussed below. When the feast of a saint falls on the leap day, it is observed on the 28th of non-leap years.

In spite of New Calendarist and pro-Gregorian claims, the Julian calendar is better suited to scientific applications. The scientific definition of a year is 365.25 days and astronomical calculations are performed in a form known as Julian Day.

Gregorian reforms

Main article: Gregorian Calendar.

Origins

Over centuries the Julian calendar drifts relative to the heavenly bodies, resulting in the seasons no longer changing on the same dates. This means that the vernal equinox defined in March for calculating the date of Pascha is no longer a physical equinox, and also results in the dates for planting, etc. in agriculture moving over time. The implications are discussed in more detail in the appendix below.

The papacy decided to compensate for the drift by instituting an immediate shift in the calendar, bringing dates back in line with the historical date of the equinox, and adjusting the leap year calculation to more accurately follow the astronomical equinox.

The New Calendar

Main article: Revised Julian Calendar.

Astronomical appendix

The Copernican revolution

The Keplerian model

Some decades after the introduction of the Gregorian calendar the mathematician Kepler, working with measurements made by Brahe, developed the first heliocentric model capable of actually predicting the positions of planets—but not the moon—with any appreciable accuracy. He achieved this by replacing the idealized circular motions of Copernicus with ellipses, where the planets speed up when closer to the sun and slow down when further away. The variation in speed is such that the line between the planet and the sun traces out an equal area over an equal length of time.

Keplerian motion still forms the basis of more complex predictions by employing the method of perturbations mentioned below.

Classical physics

The relativistic model

The theory of relativity was primarily developed by several physicists, notably Poincaré and Lorenz but is commonly attributed to Einstein (who made little initial contribution). Further development into general relativity was mostly by Einstein and possibly Hamilton, whose work was only available to Einstein.

Relativity has little bearing on the calendar. The difference with Newtonian theory is tiny and motions of the planets can be corrected for with a small perturbation (see below), effectively causing them to behave as if there were an inverse cube law term in addition to the inverse square law of Newtonian gravity. Drifts of arcseconds or fractional arcseconds per century will eventually add up but it does not seem likely that the thousands of centuries needed will pass before Judgement Day, so that they might make a day's difference to the calendar.

However, what relativity does assume is that no place in the universe is special and that no direction is different from any other. (See #isotropy below.) Because of this, under contemporary understandings of physics it does not matter whether or not we consider the earth to be the center of the universe. It is perfectly valid to perform a coordinate transformation and treat the modern view of the solar system as geocentric.

Perturbative models

The method of perturbations is commonly used to predict the long-term behavior of astronomical systems. This involves first performing a clever coordinate transform. A baseline motion is thus established—probably a Keplerian orbit—and other effects are studied as deviations from this baseline. For example if one wishes to study how the motion of Saturn is influenced by Jupiter, the gravitational pull from Jupiter may be modeled as a perturbation of the Keplerian motion of Saturn. The result produced is roughly correct on average over an orbit, so the trend over many years is made apparent. Although we can say in some sense that the results are averaged out and not exactly correct for a specific date, the effects so studied are small and it would not normally matter.

Perturbation is critical for understanding the motion of the moon because it is subject to unusually large and difficult to calculate perturbations. Hence it impacts the date of Easter. Providentially we do not need to perform these calculations; an accurate understanding of the moon's motion from first principles would have required the ancient fathers to have understood Newtonian physics and have had access to advanced computers. Instead the net result is that complex interactions correspond to relatively simpler perturbations, which result in even simpler precessional motions, which can be predicted centuries in advance without complex tools—as has been done for millennia.

Nevertheless the ancients should be credited for developing methods so advanced as to even predict eclipses. This is often glossed over in modern education and was more difficult than it is made to sound.

Computational models

Assuming one has access to a modern computer, the most accurate way to predict the movements of astronomical bodies is not to assume Keplerian orbits with small perturbations, but rather to calculate the motions directly from Newtonian (or if necessary relativistic) theory. This means solving the N-body problem, which has no closed form solution in the general case. Instead the forces must be integrated over time.

It is simple enough:

• The starting positions and velocities are known.
• Forces due to gravity are calculated from positions.
• A small step is taken forward in time, the positions being adjusted based on the velocities and the velocities being adjusted based on the forces.
• The procedure is repeated for the new state.

Since computers have finite precision the calculations must be done with a finite number of time steps, each step introduces a small error which can compound over time. More advanced methods introduce smaller errors than naive Euler integration.

Modeling limitations

Chaos theory

Chaos theory is a field of mathematics which studies unpredictability. In chaotic systems there is a compounding of errors over time, which makes it impossible to predict the future even if those systems are behaving totally in accordance with predictable rules.

As an analogy, suppose that you are driving a car in a straight line at exactly 100 km/h on an infinite flat plane. After an hour you are 100 km away, and after 100 hours, 10 000. If there is an uncertainty of 1% in your speed, then after an hour you would be 99 to 101 km away, an error of 1%. After a million hours you would be 99-101 million kilometers away, which is still an error of 1%. If instead you drove on the equator of a spherical earth with a circumference of 40 000 km, an error of 1% would still result in you being after an hour 99 to 101 km away, or 1%. After 400 hours you would be back where you started, except that you could be ahead or behind by 400 km. After 800 h you would have completed two revolutions and could be ahead or behind by 800 km. After 20 000 hours you would have completed 50 revolutions, and now you could be ahead or behind by up to 20 000 km. However, 20 000 km is the distance to the opposite end of the earth. This means that your distance that you could be ahead or behind have met, and you could be anywhere in the system (equator of the earth in this case). This concept is called Lyapunov time, and it limits the precision with which astronomical calculations can be performed into the future because even arbitrarily small measurement errors result in eventually zero knowledge of where objects will end up.

Lyapunov time for the positions of the planets in their orbits (in the Newtonian model) is relatively long—on the order of a billion years. For other motions it can be much shorter. Regardless, systems behave unpredictably when considered over several Lyapunov times. Studying it by Monte Carlo simulation, the most likely scenario for a metastable system such as the solar system would be that, after several Lyapunov times, the system would violently rearrange into a new metastable configuration, typically by way of flinging planets into outer space and rearranging the orbits of those that remain. The Church believes in Providence rather than chance, so such a scenario does not frighten her. It does however do two things: it places an absolute upper limit on how long any calendar can predict the seasons and it undermines Darwinian natural history. (Newtonian physics works mathematically the same when calculated forwards or backwards in time, and would in any case apply if you started with the solar system several billion years ago and calculated forwards until today. We would not expect to survive under materialistic assumptions.)

As an aside, the Lyapunov time for weather prediction is very roughly a week. If you ever wondered why you feel like you cannot get useful predictions, that is why.